=== n = 4 point d* = ['3/50', '3/50', '3/50', '171/1000']
Q1 = 110937519/1000000000000 = 0.000110937519
Q2 = 254677515836793921/2000000000000000000000000 = 1.2733875791839697e-07
Q1 - Q2 = 0.0001108101802420816  >0: True
Q1 - 2*Q2 (if Q2 were defined without the factor 1/2): 0.0001106828414841632 >0: True
r1^2 = 1-4a = 19/25   r4^2 = 1-4b = 79/250
(9 r1^2 - 4 - r4^2)/4 = 631/1000 >0: True
its square = 398161/1000000  vs r4^2 = 79/250  square > r4^2: True
=> 3*r1 - r4 > 2 exactly, i.e. polygon inequality f(d4) <= f(d1)+f(d2)+f(d3) FAILS
float check: 3 r1 - r4 = 2.053201 (> 2)
float: f(d4) = 0.218931, 3 f(d1) = 0.192330

=== zero-padded points for n = 5..10 (exact) ===
n=5: Q1=3.893907e-05  Q2=3.243e-14  Q1>Q2 (and >2Q2): True ; polygon at index 4 violated (same r-inequality): True
n=6: Q1=1.366761e-05  Q2=2.103e-27  Q1>Q2 (and >2Q2): True ; polygon at index 4 violated (same r-inequality): True
n=7: Q1=4.797332e-06  Q2=8.849e-54  Q1>Q2 (and >2Q2): True ; polygon at index 4 violated (same r-inequality): True
n=8: Q1=1.683864e-06  Q2=1.566e-106  Q1>Q2 (and >2Q2): True ; polygon at index 4 violated (same r-inequality): True
n=9: Q1=5.910361e-07  Q2=4.905e-212  Q1>Q2 (and >2Q2): True ; polygon at index 4 violated (same r-inequality): True
n=10: Q1=2.074537e-07  Q2=0.000e+00  Q1>Q2 (and >2Q2): True ; polygon at index 4 violated (same r-inequality): True

=== second, all-rational certificate: d' = (9/100, 9/100, 9/100, 6/25) ===
d' = ['9/100', '9/100', '9/100', '6/25']  lambda = f(d') = ['1/10', '1/10', '1/10', '2/5']
Q1 = 107811/100000000 = 0.00107811 ; Q2 = 2368521/1280000000000 = 1.85040703125e-06 ; Q1 > Q2: True ; Q1 > 2 Q2: True
polygon at index 4: lambda_4 = 2/5 > 3/10 = lambda_1+lambda_2+lambda_3 -> violated: True
boundary tensor A: |A| = 1.000000000000, Gram determinants = [0.09 0.09 0.09 0.21]
ALL EXACT CHECKS PASSED
